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[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
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[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 112: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 112: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
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[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 236: fputs() expects parameter 1 to be resource, bool given
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[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 280: copy(e580ddba0e6f13ccfbc9654e938b2da9.png): failed to open stream: No such file or directory
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[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 280: copy(e580ddba0e6f13ccfbc9654e938b2da9.png): failed to open stream: No such file or directory
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[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 297: unlink([ROOT]/latexrender//tmp/e580ddba0e6f13ccfbc9654e938b2da9.tex): No such file or directory
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[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 235: fopen([ROOT]/latexrender//tmp/e580ddba0e6f13ccfbc9654e938b2da9.tex): failed to open stream: No such file or directory
[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 236: fputs() expects parameter 1 to be resource, bool given
[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 237: fclose() expects parameter 1 to be resource, bool given
[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 280: copy(e580ddba0e6f13ccfbc9654e938b2da9.png): failed to open stream: No such file or directory
[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 281: chmod(): No such file or directory
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[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 297: unlink([ROOT]/latexrender//tmp/e580ddba0e6f13ccfbc9654e938b2da9.tex): No such file or directory
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[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 301: unlink([ROOT]/latexrender//tmp/e580ddba0e6f13ccfbc9654e938b2da9.ps): No such file or directory
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[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 235: fopen([ROOT]/latexrender//tmp/e580ddba0e6f13ccfbc9654e938b2da9.tex): failed to open stream: No such file or directory
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forum.virtualchallengemeets.com • View topic - POW: Which Decimals End in What

POW: Which Decimals End in What

What could be so hard. It's a multiple-choice test for Pete's sake.

POW: Which Decimals End in What

Postby AuSmith » Fri Apr 06, 2007 7:13 pm

(1) What are the last two digits of [unparseable or potentially dangerous latex formula], with 2007 8's?

(2) Find all the perfect squares whose last 4 digits are 2001.
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Postby Quelloquialism » Fri Apr 06, 2007 10:02 pm

Last edited by Quelloquialism on Fri Apr 06, 2007 11:27 pm, edited 1 time in total.
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8^8^8^8...

Postby collegebookworm » Fri Apr 06, 2007 10:07 pm

**edit: misread
**re-edited: corrected
first, we must find the pattern for 8^8^8^8....

notice that the pattern for 8^n, where n is any integer, starting with 1, follows this 20-cycle pattern for the last 2 digits:
08,64,12,96,68,44,52,16,28,24,92,36,88,04,32,56,48,84,72,76

to find 8^n for any n, therefore, we must find the value of n mod 20; we can thus conclude that we need the last two digits of n to find the last two digits of 8^n.

We can now find the last two digits of 8^8. Since 8 mod 20 is 8, the last two digits are 16. Now we can find the last two digits of 8^8^8 by the same logic (16th value in the pattern: 56). We can do this again (56 mod 20 = 16; 16th value in the pattern is 56) and we find that we have reached a final resting place for the last two digits: 56.
Please inform me of any errors I have made.
(NS/CA/MA/CS/SC) "**" indicates TMSCA State
08 HI(LO): 142?-**(88-D2)/224-D2(17x-**)/320-D2(202-**)/248?-D2(206-**)/210-D2(17x-NovA+?)
08 D2: 88/224/320/210/248

Next meet: Texas Tech (Region, 4/11-12)
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Postby Quelloquialism » Fri Apr 06, 2007 11:51 pm

I was getting (via Java brute-force) that the cycle was (16, 96, 36, 56) on the first problem. Not going to try a proof, though...and for all I know, my program was wrong, haha.
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Re: POW: Which Decimals End in What

Postby stupidityismygam » Fri Apr 06, 2007 11:58 pm

Last edited by stupidityismygam on Sat Apr 07, 2007 12:23 am, edited 1 time in total.
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Postby Quelloquialism » Sat Apr 07, 2007 12:14 am

My brute force was returning 96, I think...
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Postby stupidityismygam » Sat Apr 07, 2007 12:17 am

oh wait...i made a mistake....i see it....
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Postby Quelloquialism » Sat Apr 07, 2007 12:23 am

According to Wikipedia, Fermat's Little Theorem only applies when the number is of the form [unparseable or potentially dangerous latex formula]...the exponent isn't p here. Or something. Or maybe you're applying something else. I don't know...haha.
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Postby stupidityismygam » Sat Apr 07, 2007 12:24 am

yeah i made a bad assumption, i messed up my fermats....dang...
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Postby AuSmith » Sat Apr 07, 2007 1:15 am

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Re: 8^8^8^8...

Postby AuSmith » Sat Apr 07, 2007 1:22 am

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Postby makashi » Sat Apr 07, 2007 4:37 am

Let [unparseable or potentially dangerous latex formula] be the Euler Toitent Function. Then [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula]. Recall that if [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] are relatively prime, then [unparseable or potentially dangerous latex formula]. Notably, [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula]. Hence, [unparseable or potentially dangerous latex formula] Since it is obviously divisible by [unparseable or potentially dangerous latex formula], then [unparseable or potentially dangerous latex formula] by the Chinese Remainder Theorem. Thus, [unparseable or potentially dangerous latex formula]. Since [unparseable or potentially dangerous latex formula] then [unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
By the Chinese Remainder Theorem again, [unparseable or potentially dangerous latex formula], so the last two digits are [unparseable or potentially dangerous latex formula].
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Postby AuSmith » Sat Apr 07, 2007 2:18 pm

makashi's solution was more what I was looking for. I'll give mine.

[unparseable or potentially dangerous latex formula], and [unparseable or potentially dangerous latex formula]. By CRT (not cathode ray tube), [unparseable or potentially dangerous latex formula].

It follows that [unparseable or potentially dangerous latex formula]. Again, since [unparseable or potentially dangerous latex formula], [unparseable or potentially dangerous latex formula].

Finally, [unparseable or potentially dangerous latex formula] and all further numbers in the sequence are of the form [unparseable or potentially dangerous latex formula] where [unparseable or potentially dangerous latex formula], so the last two digits fix on [unparseable or potentially dangerous latex formula].
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Postby AuSmith » Sat Apr 07, 2007 3:07 pm

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Postby Quelloquialism » Sat Apr 07, 2007 4:04 pm

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Postby Quelloquialism » Wed Apr 11, 2007 10:59 pm

Any chance somebody will post a decent solution to this? It's beyond me.
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Postby AuSmith » Wed Apr 11, 2007 11:45 pm

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